On the mean square of the divisor function in short intervals

Aleksandar Ivić[1]

  • [1] Katedra Matematike RGF-a Universitet u Beogradu, Đušina 7 11000 Beograd, Serbia

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 251-261
  • ISSN: 1246-7405

Abstract

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We provide upper bounds for the mean square integral ∫ X 2 X 𝔻 k ( x + h ) - 𝔻 k ( x ) 2 d x , where h = h ( X ) ≫ 1 , h = o ( x ) as X → ∞ and h lies in a suitable range. For k ≥ 2 a fixed integer, 𝔻 k ( x ) is the error term in the asymptotic formula for the summatory function of the divisor function d k ( n ) , generated by ζ k ( s ) .

How to cite

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Ivić, Aleksandar. "On the mean square of the divisor function in short intervals." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 251-261. <http://eudml.org/doc/10879>.

@article{Ivić2009,
abstract = {We provide upper bounds for the mean square integral\[ \int \_X^\{2X\}\left(\mathbb\{D\}\_k(x+h) - \mathbb\{D\}\_k(x)\right)^2\,\{\textrm\{d\}\}x, \]where $h = h(X)\gg 1,\; h = o(x)\; \{\rm \{as\}\}\;X\rightarrow \infty $ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, $\mathbb\{D\}_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, generated by $\zeta ^k(s)$.},
affiliation = {Katedra Matematike RGF-a Universitet u Beogradu, Đušina 7 11000 Beograd, Serbia},
author = {Ivić, Aleksandar},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {mean-square; divisor functions; short intervals},
language = {eng},
number = {2},
pages = {251-261},
publisher = {Université Bordeaux 1},
title = {On the mean square of the divisor function in short intervals},
url = {http://eudml.org/doc/10879},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Ivić, Aleksandar
TI - On the mean square of the divisor function in short intervals
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 251
EP - 261
AB - We provide upper bounds for the mean square integral\[ \int _X^{2X}\left(\mathbb{D}_k(x+h) - \mathbb{D}_k(x)\right)^2\,{\textrm{d}}x, \]where $h = h(X)\gg 1,\; h = o(x)\; {\rm {as}}\;X\rightarrow \infty $ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, $\mathbb{D}_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, generated by $\zeta ^k(s)$.
LA - eng
KW - mean-square; divisor functions; short intervals
UR - http://eudml.org/doc/10879
ER -

References

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  1. G. Coppola, S. Salerno, On the symmetry of the divisor function in almost all short intervals. Acta Arith. 113(2004), 189–201. Zbl1122.11062MR2049565
  2. A. Ivić, The Riemann zeta-function. John Wiley & Sons, New York, 1985 (2nd ed., Dover, Mineola, N.Y., 2003). Zbl0556.10026MR792089
  3. A. Ivić, On the divisor function and the Riemann zeta-function in short intervals. To appear in the Ramanujan Journal, see arXiv:0707.1756. Zbl1226.11086
  4. M. Jutila, On the divisor problem for short intervals. Ann. Univer. Turkuensis Ser. A I 186 (1984), 23–30. Zbl0536.10032MR748516
  5. P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161–170. Zbl0412.10030MR552470
  6. E.C. Titchmarsh, The theory of the Riemann zeta-function (2nd ed.). University Press, Oxford, 1986. Zbl0601.10026MR882550
  7. W. Zhang, On the divisor problem. Kexue Tongbao 33 (1988), 1484–1485. MR969977

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